Prove Theorems About Perpendicular Lines Section 3 – 6 Prove Theorems About Perpendicular Lines

Theorem 3.8 If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. If 1 = 2, then g h. g 1 2 h

Theorem 3.9 If two lines are perpendicular, then they intersect to form four right angles. If a b, then 1, 2, 3, & 4 are right angles. b 1 2 a 3 4

Theorem 3.10 If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. If BA BC, then 1 and 2 are complementary. A 1 2 B C

Theorem 3.11 If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. If h k and j h, then j k. j h k

Theorem 3.12 In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. If m p and n p, then m n. m n p

Example 1 Explain how you would show that m || n . First: Find x: Since x° and x° form a linear pair we can say: x ° = 90 ° by Theorem 3.8 Therefore Second: If and then by Theorem 3.11

Example 2 2x – 9 + x = 90 3x – 9 = 90 +9 +9 3x = 99 3 3 x = 33 In the diagram, . Find the value of x 2x – 9 + x = 90 3x – 9 = 90 +9 +9 3x = 99 3 3 x = 33

Example 2 Prove that if 1 and 2 are complementary, then BA BC. ˜ Given: 1 & 2 are comp. Prove: BA BC. C D 2 1 A B Statement Reasons 1. 1 & 2 are comp. Given Def. of comp. ‘s 2. m 1 + m 2 = 90◦ 3. m ABC = m 1 + m 2 Angle Add. Post. Subst. Prop. of = 4. m ABC = 90◦ 5. ABC is a rt. Def. of rt. 6. BA BC Def. of lines

Homework Section 3-6 Page 194 – 197 5 – 11, 15 – 19, 21, 26, 39 - 43